Finally, I began questioning the kid. What was her name? Where were
her parents? And most important, what was she doing outside in this
horrible weather? She replied to none of that. In fact, she didn’t
utter a single word; all she did was to stare at me with her uncanny
eyes. Luckily, I obtained that information by other means.

That night, after the kid had gone to sleep, I found a letter in her
backpack.

Finally, I began questioning the kid. What was her name? Where were
her parents? And most important, what was she doing outside in this
horrible weather? She replied to none of that. In fact, she didn’t
utter a single word; all she did was to stare at me with her uncanny
eyes.

Luckily, I obtained that information by other means.

That night, after the kid had gone to sleep, I found a letter in her
backpack.

Finally, I began questioning the kid. What was her name? Where were
her parents? And most important, what was she doing outside in this
horrible weather? She replied to none of that. In fact, she didn’t
utter a single word; all she did was to stare at me with her uncanny
eyes.

Luckily, I obtained that information by other means. That night, after the kid had gone to sleep, I found a letter in her
backpack.

2 Answers
2

Your sentence implies that action took place which the reader is not seeing (the "other means"). Wherever that sentence is, that's where/when you're placing this off-screen action.

However, because the action of finding the letter physically happens in the next paragraph (that is, it's not off-screen), it has to be the idea or the decision to search her backpack which is the Other Means in that sentence.

In the first example, the Other Means sounds like an idea which
occurs to the narrator while s/he's questioning the girl.

In the second example, the Other Means is being presented as a
narrative device: this is the narrator explaining something to the
reader as though we were in the room with him/her, having an actual
conversation about these events.

In the third example, it sounds like the narrator only came up with
the idea to search the girl's backpack after she'd fallen asleep.

Which one works better is simply a matter of when you want the idea to occur to the narrator.

As far as tone goes, I think the first and third are equivalent. In the 2nd case, putting the sentence in a paragraph of its own adds emphasis to the statement. Also in this particular case the statement is vague which hints to the reader that the details are forthcoming which is a trick often used to build suspense.